The Paradox of Twins Described in a Three-dimensional Space-ime Frame Tower Chen E_mail: chen@uguam.uog.edu Division of Mahemaical Sciences Universiy of Guam, USA Zeon Chen E_mail: zeon_chen@yahoo.com Absrac : Afer raveling from ouer space, he elder win looks younger han his broher waiing on he earh based on he calculaion from he formula of Special Relaiviy. There is no absolue reference coordinae in he universe. Theoreically, he younger win could describe he moion of his broher flying forward, vice versa, he elder win could describe he moion of his broher flying backward. If he younger win fel his broher looking younger, he older win should also fel his broher looking younger. I is hard o comprehend ha in realiy he elder win acually looks younger han he younger win. The moion of an objec described in a hree-dimension space-ime frame by embedding ime ino space could be drawn using he graph-command of MATLAB. I leads a beer undersanding he paradox of wins based on he calculaion from he graphic mehod in he 3-d s- frame. 1. Inroducion In classical physics, he conceps of ime and space are absolue. We can alk abou space wihou specifying ime; for example, measuring he lengh of an objec in geomery wihou referring o ime. We can also alk abou ime wihou specifying space; for example, describing wo evens happening simulaneously wihou referring he locaion of an observer. Since space and ime can be separaed, space and ime are independen. We use posiion and ime o describe he moion of an objec where posiion is a funcion of ime. Alhough space and ime are discussed ogeher, here is no consrain binding space and ime. Space is composed of hree independen dimensions, and ime is composed of one independen dimension in classical physics. For example: The movemen of an objec in a space wih sec. a ( m, m, m), 1 sec. a (.5 m, 1.5 m, 1 m), 2 sec. a (1 m, 3 m, 2 m), 3 sec. a (1.5 m, 4.5 m, 3 m), and...ec. can be decomposed ino ( sec., m), (1 sec.,.5 m), (2 sec. 1 m), (3 sec., 1.5m), forward or backward along x-axis; ( sec., m), (1 sec., 1.5 m), (2 sec. 3 m), (3 sec., 4.5 m), righward or lefward along y-axis; ( sec., m), (1 sec., 1 m), (2 sec. 2 m), (3 sec., 3 m), upward or downward along
z-axis. The moion of his objec can be described in he Fig.1a, Fig.1b, and Fig.1c. (sec) (sec) (sec) x (m) y (m) z (m) Fig. 1a: The moion of he objec is decomposed along x-axis. Fig. 1b: The moion of he objec is decomposed along y-axis. Fig. 1c: The moion of he objec is decomposed along z-axis. Einsein demonsraed ha space and ime are no separable by using he experimen of emiing ligh from he middle of a car in a moving rain [1]. An observer on he rain sees he wo pulses hi he rear and fron walls of he car simulaneously. An observer on he road sees one pulse hi he rear wall before he oher hi he fron. This experimen shows ha we canno alk abou he ime of an even wihou linking i o he posiion of an observer in he space. Similarly, we canno alk abou he posiion of an objec in space wihou linking i o ime. This inseparabiliy of space and ime shows ha space and ime are dependan. Einsein furher derived Lorenz ransformaion. Frame S moves o he righ wih he velociy, v, wih respec o Frame S. A he poin where he origins of boh frmes coincide, le = =. A his ime, a beam of ligh was fired from he origin O. The beam reaches a receiver a he poin P in Frame S. In he Fig.2, he coordinaes of poin P are (x, y, z, ) wih respec o Frame S, where he beam ravels along he pah O P and he coordinaes of poin P are (x, y, z, ) wih respec o Frame S. y y S S v y S y S v P = = O O O O Fig.2: When O and O coincide, sends a beam from O and receives he beam a he poin P in Frame S. The coordinaes of P are (x,y,z, ) in Frame S and (x,y,z,) in Frame S.
The velociy of ligh measured from Frame S will be 2 2 2 x + y + z c = O P / = (1) and he velociy of ligh measured from Frame S will be 2 2 2 x + y + z c = OP / = (2) Based on he principle of he invariance of speed of ligh, he velociy of ligh measured from wo differen inerial frames remains he same. Minkowski adoped a four-dimensional space-ime (4-d s-) frame shown as Fig.3 o describe he moion of an objec. There are hree independen dimensions in space (x-axis, y-axis, z-axis), which are perpendicular o one anoher, and one independen ime dimension (-axis), which is perpendicular o all hree dimensions in space [2]. To reveal indirecly he dependency of space and ime, he added a consrain by combining Eq.(1) and Eq.(2) 2 2 2 2 2 2 2 2 x + y + z ( c) = x + y + z ( c) = cons. (3) The consrain is called he invariance of an even inerval beween wo inerial frames. y y v x x z z Fig.3: Two 4-d s- inerial frames were consruced by Minkowski o describe he moion of an objec moving in space. The way o describe he moion of an objec in a 4-d s- frame is he same as in frame of classical physics excep he consrain binding space and ime. Wih his addiional consrain, i raieses up he difficuly of finding mahemaical soluions for coordinae ransformaion problem beween wo differen frames. 2. Time Dilaion and Lengh Conracion Before discussing he paradox of wins, we need review ime dilaion and lengh conracion in Special Relaiviy. Les define some erms firs. If wo frames have a consan relaive velociy beween hem, wo frames are said o be inerial frames o one anoher. For his discussion, we will have a rain passing by a saion plaform a consan velociy. Theoreically,
we are allowed o choose any one frame of he wo frames o be he saionary inerial frame and he oher frame o be he moving inerial frame. For convenience, we consruc a saionary frame S on he plaform and a moving frame S on he rain. S O v O o o l=v o S O l o =v Fig.4: A saionary rod, laid along he side of he plaform, by a moving rain. In Fig.4, a rod is laid along he side of he saion plaform. There is an observer on he plaform and anoher observer on he rain and boh measure he rod s lengh using a sensor aached o he fron of he rain, i.e. he origin O of he moving frame S. The lengh of he rod as measured by an observer in he saionary frame S, is defined as proper lengh, l, while he lengh of he rod as measured by an observer in he moving frame S, is defined as regular lengh, l. When he sensor ouches he lef edge of he rod, he ime is recorded as for boh observers. When he sensor ouches he righ edge of he rod, he ime is recorded for he observer in he saionary frame S and for he observer in he moving frame S. The even where he sensor moves from one end of he rod o he oher can be described by he wo differen observers. This even occurs a he same locaion for he observer in he moving frame S, hen he period of he even as measured by his observer is defined as he proper ime,. This even happens a differen locaions for he observer in he saionary frame S, hen he period of he even measured by his observer is defined as he regular ime,. The proper lengh of he rod is calculaed by muliplying he rain s velociy by regular ime, l = v, (4) and regular lengh is calculaed by muliplying he rain s velociy by proper ime, l = v. (5)
O S v o O O r=c θ h=r =c o S O l o =v Fig.5: The heigh of he ceiling is adjusable. The pulse of ligh reaches he ceiling a he same ime as he sensor ouches he righ side of he rod. A he same ime he sensor ouches he lef end of he rod, he observer in he moving frame S sends a pulse of ligh owards he ceiling of he car, where a mirror is placed. To he observer in he moving frame S, he ligh ravels verically up owards he ceiling and is hen refleced verically down. The ceiling heigh of he boxcar is adjusable, such ha he pulse of ligh reaches he ceiling a he same ime ha he sensor reaches he righ end of he rod. In Fig.5, if i akes he proper ime for ligh o reach he ceiling hen he heigh of ceiling is equal o he disance raveled by ligh is h r = c =, as measured by he observer in he moving frame O. To he observer in he saionary frame O, he ligh ravels diagonally upwards o he ceiling and is hen refleced diagonally downwards. If i akes he ime for ligh o reach he ceiling hen he disance raveled by ligh on each diagonal leg is r = c, as measured by he observer in he saionary frame O, where h From Fig.5, we can derived he following propery for θ, where sin 2 2 2 2 = r l = ( c) ( v) (6) 2 2 2 2 2 θ = h / r = r l / r = ( c) ( v) / c = 1 ( v / c) (7) From he previous discussion, we know ha r = c and h r = c =, hen sinθ = h / r = r / r = c / c / and = =. (8) 2 / 1 ( v / c) Therefore,. This equaion shows ha he regular ime,, is larger han or equal o
he proper ime,. This resul says ha he ime inerval measured by he observer in he saionary frame is longer han ha measured by he observer in he moving frame. This difference is referred o as ime dilaion. Since sinθ = /, hen sinθ = = l and / = v / v l / l = l. (9) 2 1 ( v / c) Therefore, l l. This equaion shows ha he regular lengh, l, is less han or equal o he proper lengh, l. This resul says ha he lengh of a rod measured by he observer in he moving frame is shorer han ha measured by he observer in he saionary frame. This difference is referred o as lengh conracion. 3. Consrucing a 3-d s- Coordinae Sysem Tradiionally speaking, lengh and ime are fundamenal quaniies and velociy is a derived quaniy. As a maer of fac, if no objec in he universe exhibis any moion; we would have no concep of space or ime. Because here was moion of an objec, he occupaion of space could be observed and he order of ime (he process from he beginning o he ending of an even) could be recorded. Hence, velociy should be reaed as a fundamenal quaniy and lengh and ime should be reaed as derived quaniies. Periodic moions are he mos regular ype of moion and ligh waves are he seadies ype of periodic moion. The velociy of ligh is consan for all observers and independen from wavelenghs. Hence, a hree-dimensional space-ime (3-d s-) frame is consruced wih waves of ligh moving along x-axis, y-axis, z-axis. Based on Einsein s hough experimen abou wo simulaneous evens on a moving rain and invarian velociy of ligh, space and ime are dependen. We posulae ha here is a consrain in a 3-d s- frame by embedding ime ino space for describing he moion of an objec [3]. The waves of ligh along he x-axis, y-axis, z-axis are chosen as he foundaion o consruc a 3-d s- frame. The unis of he dimensions of space and ime are consrained by he velociy of ligh: λ / T = λ / T = c (1) The consrain is he invariance of he raio of a uni of lengh o a uni of ime. Waves along x-axis, y-axis, z-axis are chosen from he same ype of ligh waves, because hey have he same wavelengh and period. MATLAB has exensive faciliies for displaying vecors and marices as graphs. I includes high-level funcion for wo-dimensional daa and hree-dimensional daa o describe he moion of an objec moving in space in graphs.
z (2λ,2T) (1λ,1T) 2T (1λ,1T) O (1λ,1T) y (2λ,2T) O 1T 1λ 2λ (2λ,2T) x Fig.6a: A 3-d s- frame is composed of x,y,z axes which are perpendicular o one anoher and spherical surfaces of differen radii which represens he proceeding of ime. Fig.6b: For any dimension of a 3-d s- frame, λ is chosen o be he uni of axis of space, and T is chosen o be he uni of radii of ime such ha hey saisfy he consrain λ/t= c. Spherical surfaces of differen radii are used o represen he proceeding of ime having he uni of a wave period, T, of ligh. Three axes are used o represen he disance in space, each wih he uni of a wavelengh, λ, of ligh. The coordinaes of he inerceps among spheres of ime and hree dimensions of space are (1λ,1T) x, (2λ,2T) x,... (nλ,nt) x on he x-axis, (1λ,1T) y, (2λ,2T) y,... (nλ,nt) y,... (nλ,nt) y on he y-axis, (1λ,1T) z, (2λ,2T) z,... (nλ,nt) z on he z-axis. The 3-d s- frame is shown on Fig.6. This 3-d s- frame is anoher represenaion of space-ime coordinae sysem. A 3-d s- frame no only reveals direc dependency of space and ime bu also has he simples consrain. Though he space coordinaes are bi-direcional, ime only has one ougoing direcion in his 3-d s- frame. Since a 3-d s- frame by embedding ime ino space can be comprehended easily, i is chosen as he coordinae sysem o describe moions of an objec in his paper. 4. The Moions of he Elder Twin Described in Two 3-d s- Frames The younger win was in frame O and he elder win was in he frame O. If he elder win in he frame O flied o he sar from he earh wih 8% of he velociy of ligh, hen he velociy, v, is (4/5)c. Afer reaching he sar, hen he elder win on he frame O flied back wih he same velociy o mee he younger win in he frame O saying on he earh. Assuming he ime of he velociy acceleraed from zero o (4/5)c or deceleraed from (4/5)c o zero is very shor. There is no absolue moion in he universe, so here was only relaive moion beween he frame O and he frame O. Because of he reversed direcion of moion due o he frame O flying back, only he frame O can be reaed as a saionary inerial frame and only he frame O can be reaed as a moving inerial frame [4].
B 1T 9T 8T 7T 6T 5T 4T 3T 2T A 1T 1λ 2λ 3λ 4λ 5λ 6λ 7λ 8λ 9λ 1λ O O x From Fig.7, he moion of he elder win in he frame O flying o he sar wih he velociy v = (4 / 5)c could be described as he line OA by an observer in a 3-d s- frame O and he line O A by an observer in he 3-d s- frame O. The raio of OO /OA is always 4/5 for any righ riangle similar o he righ riangle OAO. The ime,, which he ook, measured by he observer in he frame O is called proper ime. The moion of he elder win in he frame O flying o he sar could be described as he line OA by an observer in a 3-d s- frame O. The ime,, which he ook, measured by he observer in he frame O is called regular ime. If he elder win raveled wih 8% velociy of ligh, hen O A could be calculaed as 3 by Eq.(11) from he righ riangle OAO of he graph where OA = 5, OO = 4. Fig. 7: The moion of he elder win on he frame O flied from he earh o he sar wih he velociy v=(4/5)c. was described as he line OA by an observer on he frame O. The moion of he elder win was described as he line O A by an observer on he frame O. 2 2 2 2 O A = OA OO = 5 4 = 3 (11) If i ook he period of proper ime, measured from he observer in he frame O, for he elder win flied from he earh o he sar, hen i ook he period of relaed ime, measured from he observer in he frame O. The period of regular ime and he period of proper ime have he following relaion: : = 3 : 5. (12) I can be rewrien as = (3/ 5). (13) For example: If i ook 1 years for he elder win o fly from he earh o he sar measured on he frame O, hen i ook 6 years measured on he frame O. This resul using graphic mehod in a 3-d s- frame is he same resul as calculaed by Eq.(8) of Special Relaiviy.
4.1 The Moion of Elder Twin Described by an Observer on he Earh B 1T 9T 8T 7T 6T 5T 4T 3T 2T A 1T O 1λ 2λ 3λ 4λ 5λ 6λ 7λ 8λ 9λ 1λ Fig. 8: The moion of he elder win in he frame O flied o he sar wih a velociy v=(4/5)c was described as he line OA and afer reaching he sar, he flied back wih he same velociy o mee his broher described as he line AB in a 3-d s- frame by an observer in he frame O. From Fig.8, he moion of he elder win in he frame O could be described by he line OA flying o he sar and he line AB flying back o he earh in a 3-d s- frame by an observer in he frame O. Because he velociy of flying back was he same as he velociy of flying ou, he period ime of he round rip for he elder win was 1 years measured by he observer in he frame O. x 4.2 The Moion of he Elder Twin by an Observer on Space Sheler From Fig.9, he moion of he elder win in he frame O could be described by he line O A flying o he sar and he line AC flying back o he earh in a 3-d s- frame by an observer in he frame O. Because he velociy of flying back was same as flying ou, he round rip for he elder win was 6 years measured by he observer in he frame O. C 6T 5T 4T A 3T 2T 1T O 1λ 2λ 3λ 4λ 5λ 6λ x Fig. 9: The moion of he elder win in he frame O flied o he sar wih a velociy v=(4/5)c was described as he line O A and afer reaching he sar, he flied back wih he same velociy o mee his broher described as he line AC in a 3-d s- frame by an observer in he frame O.
5. Conclusion By he same graphic mehod using an adequae righ riangle for he frame O moving wih differen percenage of he velociy of ligh, we are able o calculae he proper ime measured by an observer in he frame O relaed o he regular ime measured by an observer in he frame O. The resuls are lised in he following able: Velociy 1%c 2%c 3%c 4%c 5%c 6%c 7%c 8%c 9%c 95%c 99%c Regular 1 1 1 1 1 1 1 1 1 1 1 ime years years years years years years years years years years years Proper 9.94 9.8 9.54 9.16 8.66 8 7.14 6 4.36 3.12 1.42 ime years years years years years years years years years years years The moion of an objec described in a 3-d s- frame leads an easier and beer undersanding ha afer raveling from ouer space. The elder win looks younger han his broher waiing on he earh based on he calculaion from he graphic mehod in he 3-d s- frame. These resuls are he same resuls calculaed from he formula of Special Relaiviy. 6. Acknowledgemens We would like o hank Bo Liu for a fruiful collaboraion. 7. References [1] A. Einsein, The Special and General Theory (Random House, Incorporaed, 1977). [2] J. J. Callahan, The Geomery of Space Time: An Inroducion o Special and General Relaiviy (Springer Verlag, 2). [3] T. Chen and Z. Chen, Moion of Paricles Described in a Three-Dimensional Space-Time Frame, he Proceedings of he 1h Asian Technology Conference in Mahemaics, December 12-16, 25. [4] T. Chen and Z. Chen, A Pair of Inerial Frames versus an Inerial Frame, Volume V(28), Number 3, he Journal of Conceps of Physics, www.uni.lodz.pl/conceps.